Evaluation of a Determinant

Q.

How do you find the exact value for $\tan \left(120°\right)?$

Q.

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha +\beta =1$and $\alpha \beta =–1$. Let$P_n={\left(\alpha \right)}^n+{\left(\beta \right)}^n,P_{n-1}=11$ and $P_{n+1}=29$ for some integer $n\ge 1$Then, the value of ${P_n}^2$ is______________.

Q. If $A=\left[\begin{array}{ccc}{a}^2& ab& ac\\ ab& b^2& bc\\ ac& bc& c^2\end{array}\right]\text{and}{a}^2+b^2+c^2=1,$ then $A^2=$

1. $A$
2. $2A$
3. $3A$
4. $4A$

Q.

If $\sin \left({\theta }_1\right)+\sin \left({\theta }_2\right)+\sin \left({\theta }_3\right)=3$, then $\cos \left({\theta }_1\right)+\cos \left({\theta }_2\right)+\cos \left({\theta }_3\right)=$

1. $3$

2. $2$

3. $1$

4. $0$

Q. Let $A$ be a $2\times 2$ matrix with det $\left(A\right)=-1$ and det $\left(\right(A+I\left)\right($Adj $\left(A\right)+I\left)\right)=4$. Then the sum of the diagonal elements of $A$ can be

Q. If A is a matrix of order 3 and |A| = 8, then |Adj A| =

1. $1$
2. $2$
3. $2^3$
4. $2^6$

Q.

If $A=\left[\begin{array}{cc}2& 3\\ 0& -1\end{array}\right]$, then the value of $\det \left(A^4\right)+\det \left[{\left(A\right)}^{10}-\mathrm{Adj}{\left(2A\right)}^{10}\right]$ is equal to

Q.

What is mean by closed bracket and open bracket in inverse trigonometric functions and function please elaborate deeply with examples

Q.

If $A=\left[\begin{array}{cc}\alpha & 2\\ \alpha & 2\end{array}\right]$ & $\left|A^3\right|=125$ then the value of $\mathit{\alpha }$ is

1. $\pm 1$

2. $\pm 2$

3. $\pm 3$

4. $\pm 5$

Q.

Let$p,q$ be two positive number such that $p+q=2$and $p^4+q^4=272$. Then $p$ and $q$ are roots of the equation:

1. $x^2–2x+2=0$

2. $x^2–2x+8=0$

3. $x^2–2x+136=0$

4. $x^2–2x+16=0$

Q. If $A=\left[\begin{array}{cc}1& \tan x\\ -\tan x& 1\end{array}\right]$, then $A^T{A}^{-1}$ is

1. $\left[\begin{array}{cc}-\cos 2x& \sin 2x\\ -\sin 2x& \cos 2x\end{array}\right]$
2. $\left[\begin{array}{cc}\cos 2x& -\sin 2x\\ \sin 2x& \cos 2x\end{array}\right]$
3. $$\left[\begin{array}{cc}\cos 2x& \cos 2x\\ \cos 2x& \sin 2x\end{array}\right]$$
4. $$\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$$

Q.

Integrate:

Sin2x/Sin5xSin3x dx

Q.

If $x+\frac{1}{x}=2\cos \theta$, then $x^n+\frac{1}{x^n}$ is equal to

1. $2\sin n\theta$

2. $2\cos n\theta$

3. $\sin \left(2^n\theta \right)$

4. $\cos \left(2^n\theta \right)$

Q.

If$A=\left[\begin{array}{cc}2& -1\\ 1& 2\end{array}\right]$then $A^2+2A-3I=$

1. $\left[\begin{array}{cc}4& -6\\ 6& 4\end{array}\right]$

2. $0$

3. $\left[\begin{array}{cc}-6& 2\\ 2& 6\end{array}\right]$

4. $5I$

Q. If $\left(\begin{array}{cc}3& y\\ x& 1\end{array}\right|=\left(\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$, where x and y are natural numbers, then possible values of x and y are:

1. x = 1, y = 8
2. x = 2 , y = 4
3. x = 4 , y = 2
4. x = 8 , y = 1

Q.

If $f\left(x\right)={\sin }^{2}x+{\sin }^2\left(x+\frac{\mathrm{\pi }}{3}\right)+\cos x\cos \left(x+\frac{\mathrm{\pi }}{3}\right)$ and $g\left(\frac{5}{4}\right)=1$, then $gof\left(x\right)$ is equal to

1. $1$

2. $-1$

3. $2$

4. $-2$

Q. If $f:S\to R$ where $S$ is the set of all non-singular matrices of order $2$ over $R$ and $f\left[\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\right]=ad-bc$, then

1. $f$ is neither one-one nor onto
2. $f$ is bijective mapping
3. $f$ is one-one but not onto
4. $f$ is onto but not one-one

Q.

Using Cofactors of elements of second row, evaluate.

Q.

If, then find the value of x.

Q.

Let $\mathrm{M}=\left[\begin{array}{ccc}0& 1& \mathrm{a}\\ 1& 2& 3\\ 3& \mathrm{b}& 1\end{array}\right]$ and $\mathrm{adj}\left(\mathrm{M}\right)=\left[\begin{array}{ccc}-1& 1& -1\\ 8& -6& 2\\ -5& 3& -1\end{array}\right]$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?

1. $\mathrm{a}+\mathrm{b}=3$

2. $\det \left(\mathrm{adj}{\mathrm{M}}^2\right)=81$

3. ${\left(\mathrm{adj}\mathrm{M}\right)}^{-1}+\mathrm{adj}{\mathrm{M}}^{-1}=-\mathrm{M}$

4. If $\mathrm{M}\left[\begin{array}{l}\mathrm{\alpha }\\ \mathrm{\beta }\\ \mathrm{\gamma }\end{array}\right]=\left[\begin{array}{l}1\\ 2\\ 3\end{array}\right]$ then $\mathrm{\alpha }-\mathrm{\beta }+\mathrm{\gamma }=3$

Q.

If $4a^2+9b^2+16c^2=2(3ab+6bc+4ca)$, where $a,b$ and $c$ are non - zero numbers, then $a,b$ and $c$ are in

1. $AP$

2. $GP$

3. $HP$

4. None of these

Q. The set of all values of $\lambda$ for which the system of linear equations $2x_1-2x_2+x_3=\lambda x_1,2x_1-3x_2+2x_3=\lambda x_2$ and $-x_1+2x_2=\lambda x_3$ has a non - tirvial solution.

1. contains more than two elements
2. is an empty set
3. is a singleton set

4. contains two elements

Q. Let $\omega \ne 1$ be a cube root of unity and S be the set of all non - singular matrices of the form $\left[\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right]$, where each of a, b and c is either $\omega$ or ${\omega }^2$. Then, the number of distinct matrices in the set S is

1. 2
2. 8
3. 6
4. 4

Q.

If $a,b,c,d$are distinct integers in $A.P$ such that $d=a^2+b^2+c^2,$ then $a+b+c+d$is

1. $0$

2. $1$

3. $2$

4. None of these

Q. Let $a_1,a_2,...,a_{10}$ be an AP with common difference $-3$ and $b_1,b_2,...,b_{10}$ be a GP with common ratio $2.$ Let $c_k=a_k+b_k,k=1,2,...,10.$ If $c_2=12$ and $c_3=13,$ then $\sum _{k=1}^{10}{c}_k$ is equal to

Q. If $p{\lambda }^4+q{\lambda }^3+r{\lambda }^2+s\lambda +t=\left|\begin{array}{ccc}{\lambda }^2+3\lambda & \lambda -1& \lambda +3\\ \lambda +1& 2-\lambda & \lambda -4\\ \lambda -3& \lambda +4& 3\lambda \end{array}\right|,$ the value of t is

1. 16
2. 18
3. 17
4. 19

Q. The value of the determinant $\left|\begin{array}{ccc}4& -6& 1\\ -1& -1& 1\\ -4& 11& -1\end{array}\right|$ is

1. -75
2. 25
3. 0
4. -25

Q. If b+c , c+a , a+b are in HP then, a/b+c , b/c+a, c/a+b are in
a)AP b)GP c)HP d)none

Q. Let $y=\left({\cot }^{-1}x\right)\left({\cot }^{-1}\right(-x\left)\right)$ and range of $y\in (0,\frac{a{\pi }^2}{b}]$, then the value of $a+b$ is

1. $2$
2. $4$
3. $5$
4. $6$

Q. $A=\left[\begin{array}{cc}1& \tan x\\ -\tan x& 1\end{array}\right]$ and $f\left(x\right)$ is defined as $f\left(x\right)=det.A^T{A}^{-1}$ then the value of $\frac{f\left(f\right(f......f\left(x\right)\left)\right))}{n\text{times}}(\text{for}n\ge 2)$ is